Unital embeddings of the Jiang--Su algebra are not unique

We show that the canonical embeddings of the Jiang–Su algebra $\mathcal{Z}$ into the universal unital free product $\mathcal{Z}{\ast }_{\mathbb{C}}\mathcal{Z}$ are not approximately unitarily equivalent. This answers one of the $99$ problems in the list of Schafhauser, Tikuisis, and White. By contrast, we prove that any two such embeddings into a simple, unital, monotracial ${\mathrm{C}}^{\ast }$-algebra with strict comparison by the unique trace are strongly asymptotically unitarily equivalent.